Interactions of three components and subcritical self-sustained amplification of disturbances in plane Poiseuille flow

A low-dimensional nonlinear model for the normal velocity $(v)$ and normal vorticity (η) disturbance development in plane Poiseuille flow is studied. The study is restricted to the interactions of a pair of oblique components of the form $ei(αx±βz)$ and the component of the form $ei2βz,$ where α and β are streamwise and spanwise wave numbers, respectively. The disturbances considered are also assumed to be highly elongated in the streamwise direction. Owing to the non-normal properties of the basic equations, the η disturbance is first transiently amplified. Then, if the Reynolds number (R) and the initial disturbance are sufficiently large, the nonlinear interactions lead to a self-sustained process of disturbance amplification at subcritical R. For large R $(R≳5000),$ the threshold disturbance amplitude scales like $R−3.$ The results also strongly indicate that the nonlinear feedback from η to $v$ is crucial for the establishment of the self-sustained process.